First slide

Applications of determinant

Question

If A and B are two square matrices such that $B = - A^{- 1} B A t h e n {(A + B)}^{2} =$

Moderate

A

0
B

$A^{2} + B^{2}$
C

$A^{2} + 2 A B + B^{2}$
D

$A + B$

Solution

$\begin{array}{l} B = A^{- 1} B A \\ \Rightarrow A B = - A (A^{- 1} B A) \\ = - B A \\ (A + B)^{2} = (A + B) (A + B) \\ = A (A + B) + B (A + B) \\ = A^{2} + A B + B A + B^{2} \\ = A^{2} + B^{2} \end{array}$

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